What Is The Range Of The Function G G G ? G ( X ) = X − 1 + 2 G(x) = \sqrt{x-1} + 2 G ( X ) = X − 1 ​ + 2 A. Y ≥ 1 Y \geq 1 Y ≥ 1 B. Y ≥ 2 Y \geq 2 Y ≥ 2 C. Y ≤ 2 Y \leq 2 Y ≤ 2 D. Y ≤ 1 Y \leq 1 Y ≤ 1

What is the Range of the Function $g$?

Understanding the Function $g(x)$

The function $g(x) = \sqrt{x-1} + 2$ is a square root function with a horizontal shift of 1 unit to the right. This means that the function is undefined when $x-1$ is negative, as the square root of a negative number is not a real number.

Determining the Domain of the Function $g(x)$

To determine the domain of the function $g(x)$, we need to find the values of $x$ for which $x-1$ is non-negative. This is because the square root of a negative number is not a real number.

x - 1 \geq 0
x \geq 1

Therefore, the domain of the function $g(x)$ is $x \geq 1$.

Determining the Range of the Function $g(x)$

To determine the range of the function $g(x)$, we need to find the set of all possible values of $g(x)$.

Let $y = g(x) = \sqrt{x-1} + 2$. We can rewrite this equation as:

y - 2 = \sqrt{x-1}
(y - 2)^2 = x - 1
x = (y - 2)^2 + 1

Since $x \geq 1$, we know that $(y - 2)^2 \geq 0$. Therefore, we can conclude that:

(y - 2)^2 \geq 0
y - 2 \geq 0
y \geq 2

This means that the range of the function $g(x)$ is $y \geq 2$.

Conclusion

In conclusion, the range of the function $g(x) = \sqrt{x-1} + 2$ is $y \geq 2$. This is because the function is undefined when $x-1$ is negative, and the square root of a negative number is not a real number. Therefore, the correct answer is:

The final answer is B. $y \geq 2$
Q&A: Understanding the Range of the Function $g$

Frequently Asked Questions

We have received several questions about the range of the function $g(x) = \sqrt{x-1} + 2$. Here are some of the most frequently asked questions and their answers:

Q: What is the range of the function $g(x)$?

A: The range of the function $g(x)$ is $y \geq 2$. This is because the function is undefined when $x-1$ is negative, and the square root of a negative number is not a real number.

Q: Why is the function $g(x)$ undefined when $x-1$ is negative?

A: The function $g(x)$ is undefined when $x-1$ is negative because the square root of a negative number is not a real number. In other words, $\sqrt{-1}$ is not a real number.

Q: How do I determine the range of a function?

A: To determine the range of a function, you need to find the set of all possible values of the function. This can be done by analyzing the function and its domain.

Q: What is the domain of the function $g(x)$?

A: The domain of the function $g(x)$ is $x \geq 1$. This is because the function is undefined when $x-1$ is negative.

Q: Can I use the function $g(x)$ to model real-world phenomena?

A: Yes, the function $g(x)$ can be used to model real-world phenomena. For example, the function can be used to model the growth of a population over time.

Q: How do I graph the function $g(x)$?

A: To graph the function $g(x)$, you can use a graphing calculator or a computer algebra system. You can also use a piece of graph paper and a pencil to graph the function by hand.

Q: What is the significance of the range of a function?

A: The range of a function is significant because it tells us the set of all possible values of the function. This can be useful in a variety of applications, such as modeling real-world phenomena or solving equations.

Q: Can I use the function $g(x)$ to solve equations?

A: Yes, the function $g(x)$ can be used to solve equations. For example, you can use the function to solve the equation $y = g(x)$.

Q: How do I use the function $g(x)$ to solve equations?

A: To use the function $g(x)$ to solve equations, you need to substitute the function into the equation and solve for the variable. For example, you can use the function to solve the equation $y = g(x)$ by substituting $g(x)$ into the equation and solving for $y$.

Q: What are some common applications of the function $g(x)$?

A: Some common applications of the function $g(x)$ include modeling population growth, modeling the growth of a chemical reaction, and modeling the growth of a physical system.

Q: Can I use the function $g(x)$ to model non-linear phenomena?

A: Yes, the function $g(x)$ can be used to model non-linear phenomena. For example, you can use the function to model the growth of a population over time.

Q: How do I use the function $g(x)$ to model non-linear phenomena?

A: To use the function $g(x)$ to model non-linear phenomena, you need to substitute the function into the equation and solve for the variable. For example, you can use the function to model the growth of a population over time by substituting $g(x)$ into the equation and solving for the population.

Q: What are some common mistakes to avoid when using the function $g(x)$?

A: Some common mistakes to avoid when using the function $g(x)$ include:

• Not checking the domain of the function
• Not checking the range of the function
• Not using the correct substitution for the function
• Not solving the equation correctly

Q: How do I avoid these common mistakes?

A: To avoid these common mistakes, you need to carefully read and understand the problem, check the domain and range of the function, and use the correct substitution for the function. You also need to solve the equation correctly and check your work.

Q: What are some resources for learning more about the function $g(x)$?

A: Some resources for learning more about the function $g(x)$ include:

• Textbooks on algebra and calculus
• Online resources such as Khan Academy and Wolfram Alpha
• Graphing calculators and computer algebra systems
• Online forums and discussion groups

Q: How do I use these resources to learn more about the function $g(x)$?

A: To use these resources to learn more about the function $g(x)$, you need to carefully read and understand the material, practice solving problems, and ask questions when you need help. You also need to use the resources to check your work and ensure that you are using the function correctly.